Finite Group Databasebeta

← #29670404

Subgroups — #29670404

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458815C211yesmaximal, characteristic, complemented
2#21458477C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445459C211yesmaximal, characteristic, complemented
5#21445352C211yesmaximal, characteristic, complemented
6#21445260C211yesmaximal, characteristic, complemented
7#21445233C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445032C211yesmaximal, characteristic, complemented
10#21444991C211yesmaximal, characteristic, complemented
11#21444877C211yesmaximal, characteristic, complemented
12#21444873C211yesmaximal, characteristic
13#21444660C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic, complemented
17#16840469C2211yescharacteristic
18#16840144—21no—
19#16840138C2211yescharacteristic
20#16840136C2211yescharacteristic, complemented
21#16840118—21no—
22#16840114C2211yescharacteristic, complemented
23#16840109C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840078—21no—
26#16840061—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic
29#16839989—21no—
30#16839984—21no—
31#16839979—21no—
32#16839974—21no—
33#16839969C2211yescharacteristic, complemented
34#16839959C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839913—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C8×Dic122C2211yescharacteristic, complemented
40C4×#11758201—21no—
41C4×#11758200C2211yescharacteristic
42#16839842—21no—
43D4×#92418C2211yescharacteristic
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200D411yescharacteristic
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758199—41no—
75C22×#92418—21no—
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519768—21no—
79#14519766—21no—
80#14519765—21no—
81#14519762—41no—
82#14519756—21no—
83#14519754—41no—
84C4×Dic122C2311yescharacteristic
85C4×D244—21no—
86D4×Dic61—21no—
87#14519739—21no—
88C2×C4⋊Dic61—21no—
89#14519710—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519700—41no—
100#14519700D411yescharacteristic
101#14519699—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61D411yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61—41no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×Dic61—41no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122C2×D411yescharacteristic
167SD16×C61—22no—
168SD16×C61—81no—
169M16×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758202—81no—
190#11758201—42no—
191#11758201—81no—
192#11758200—41no—
193#11758200—81no—
194#11758199—42no—
195#11758199—81no—
196#11758199—82no—
197#11758198—41no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206C4×Dic61—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—41no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—161no—
220C22×D61—41no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61#13512yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—161no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—81no—
257C4×D61—161no—
258Dic122—21no—
259Dic122—21no—
260Dic122—81no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19825—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—164no—
281C22×C61D4211yescharacteristic
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—41no—
296Dic61—41no—
297Dic61—81no—
298#3394—611no—
299#3361—611no—
300#3324D6111yescharacteristic, complemented
301#3295—611no—
302#1717—611no—
303#1697—611no—
304#1682—611no—
305#1678—611no—
306#1665—611no—
307#1644—611no—
308#1638—611no—
309#1620—611no—
310#1619—611no—
311#1583—611no—
312SD16⋊C8—611no—
313C2×C61—42no—
314C2×C61—81no—
315C2×C61—161no—
316C2×C61#326811yescharacteristic
317C2×C61#328211yescharacteristic
318C2×C61#328811yescharacteristic
319D61—161no—
320#559—611no—
321#556C2×D6111yescharacteristic
322#554—611no—
323D42C2×D6111yescharacteristic
324#495—1221no—
325#493—611no—
326#492—611no—
327#488—1221no—
328#487—611no—
329#486—611no—
330#486—611no—
331#486C2×D6111yescharacteristic
332#482—21no—
333#481—1221no—
334#478—1221no—
335#477—611no—
336#476—611no—
337#474C2×D6111yescharacteristic
338#468—21no—
339#465—1221no—
340#464—1221no—
341#463—1221no—
342#463—1221no—
343#462—611no—
344#460—611no—
345#460C2×D6111yescharacteristic
346#459—611no—
347#459—611no—
348#450—21no—
349#450—1221no—
350C8×Q8—611no—
351C8×Q8—611no—
352C4×SD16—611no—
353C4×SD16—1221no—
354C4×SD16C2×D6111yescharacteristic
355C4×D8—1221no—
356#436C2×D6111yescharacteristic
357C8×D4—611no—
358#418—21no—
359C2×#102—1221no—
360#335—611no—
361#334—611no—
362#333—611no—
363#332—611no—
364#331—611no—
365#331—611no—
366Q8⋊C8—611no—
367Q8⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370#322—611no—
371C61#1982511yescharacteristic, complemented, Sylow
372C22×D4—21no—
373C22×D4—21no—
374C2×Q16—1221no—
375C2×SD16—21no—
376C2×SD16—611no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16C22⋊D6111yescharacteristic
381C2×D8—1221no—
382C2×D8—1221no—
383C2×D8—2441no—
384C2×M16—21no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387C4⋊Q8C22×D6111yescharacteristic
388#127—611no—
389#127C22×D6111yescharacteristic
390#126—41no—
391#125—1221no—
392#124—21no—
393#124—1221no—
394#124—1221no—
395#123—2441no—
396C22⋊Q8—21no—
397C22⋊Q8—1221no—
398C22⋊Q8—2441no—
399#121—21no—
400#121—21no—
401#120—41no—
402C4×Q8—21no—
403C4×Q8—611no—
404C4×Q8C22×D6111yescharacteristic
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409C4×D4C22×D6111yescharacteristic
410#117—21no—
411C2×C4⋊C4—1221no—
412#107—21no—
413#107—1221no—
414#106—611no—
415#106—1221no—
416#106C22⋊D6111yescharacteristic
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6111yescharacteristic
423#103—611no—
424#103—1221no—
425#103—1221no—
426#103—2441no—
427#103C22⋊D6111yescharacteristic
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102C22⋊D6111yescharacteristic
442C22⋊C8—21no—
443C22⋊C8—1221no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C24—41no—
450C2×Q8—21no—
451C2×Q8—611no—
452C2×Q8—1221no—
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4D4×D6111yescharacteristic
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—2441no—
471C22×C4—2441no—
472Q16—4881no—
473SD16—22no—
474SD16—81no—
475SD16—4881no—
476SD16—4881no—
477SD16—2441no—
478SD16—2442no—
479D8—4881no—
480D8—4882no—
481D8—2442no—
482M16—81no—
483C2×C8—21no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—4881no—
491C2×C8C2×C22⋊D6111yescharacteristic
492C2×C8C2×C22⋊D6111yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4D4×D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C42—21no—
516C42—41no—
517C42—1221no—
518C42C23×D6111yescharacteristic
519C23—21no—
520C23—21no—
521C23—41no—
522C23—82no—
523C23—82no—
524C23—161no—
525C23—2441no—
526Q8—21no—
527Q8—21no—
528Q8—41no—
529Q8—41no—
530Q8—1221no—
531Q8—1221no—
532Q8—4881no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—81no—
541D4—82no—
542D4—82no—
543D4—4881no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548D4#1451979712yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—9761no—
561C2×C4—9761no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#1451982211yescharacteristic
566C8—21no—
567C8—21no—
568C8—41no—
569C8—1221no—
570C8—1221no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—164no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22D4×C22⋊D6111yescharacteristic
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—4881no—
596C4—2441no—
597C4—2441no—
598C4C2×#1451979711yescharacteristic
599C4#1684055811yescharacteristic
600C4#1684056611yescharacteristic
601C4#1684057311yescharacteristic
602C2—42no—
603C2—81no—
604C2—161no—
605C2—9761no—
606C2#2145735311yescharacteristic
607C2#2145752211yescharacteristic
608C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radical#21444873C2
hypercenter#3324D61
hypocenterC61#19825
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3324

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21458815C2semidirect
2. #21458477C2semidirect
3. #21457643C2semidirect
4. #21445459C2semidirect
5. #21445352C2semidirect
6. #21445260C2semidirect
7. #21445233C2semidirect
8. #21445162C2semidirect
9. #21445032C2semidirect
10. #21444991C2semidirect
11. #21444877C2semidirect
13. #21444660C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3324C2semidirect
16. #16840497C22semidirect
20. #16840136C22semidirect
22. #16840114C22semidirect
23. #16840109C22semidirect
27. #16840057C22semidirect
33. #16839969C22semidirect
38. Q8×#92418C22semidirect
39. C8×Dic122C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839483C22semidirect
300. #3324D61semidirect
371. C61#19825semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214588152
#214584772
#214576432
#214454592
#214453522
#214452602
#214452332
#214451622
#214450322
#214449912
#214448772
#214448732
#214446602
SD16⋊#924182
C61×#33242
#1982561